3.1.81 \(\int (d+e x)^{3/2} (a+b \text {sech}^{-1}(c x)) \, dx\) [81]

3.1.81.1 Optimal result

Integrand size = 18, antiderivative size = 343 \[ \int (d+e x)^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} \sqrt {1-c^2 x^2}}{15 c^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {28 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^3 \sqrt {d+e x}}-\frac {4 b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 e \sqrt {d+e x}} \]

output

2/5*(e*x+d)^(5/2)*(a+b*arcsech(c*x))/e-28/15*b*d*EllipticE(1/2*(-c*x+1)^(1 
/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(e* 
x+d)^(1/2)/c/(c*(e*x+d)/(c*d+e))^(1/2)-4/15*b*(2*c^2*d^2+e^2)*EllipticF(1/ 
2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(1/(c*x+1))^(1/2)*(c*x 
+1)^(1/2)*(c*(e*x+d)/(c*d+e))^(1/2)/c^3/(e*x+d)^(1/2)-4/5*b*d^3*EllipticPi 
(1/2*(-c*x+1)^(1/2)*2^(1/2),2,2^(1/2)*(e/(c*d+e))^(1/2))*(1/(c*x+1))^(1/2) 
*(c*x+1)^(1/2)*(c*(e*x+d)/(c*d+e))^(1/2)/e/(e*x+d)^(1/2)-4/15*b*e*(1/(c*x+ 
1))^(1/2)*(c*x+1)^(1/2)*(e*x+d)^(1/2)*(-c^2*x^2+1)^(1/2)/c^2
 

3.1.81.2 Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 21.84 (sec) , antiderivative size = 2653, normalized size of antiderivative = 7.73 \[ \int (d+e x)^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Result too large to show} \]

input

Integrate[(d + e*x)^(3/2)*(a + b*ArcSech[c*x]),x]
 

output

Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[d + e*x]*((-4*b*e)/(15*c^2) - (4*b*e*x)/(15 
*c)) + Sqrt[d + e*x]*((2*a*d^2)/(5*e) + (4*a*d*x)/5 + (2*a*e*x^2)/5) + (2* 
b*(d + e*x)^(5/2)*ArcSech[c*x])/(5*e) - (4*b*(7*c*d*e*Sqrt[(1 - c*x)/(1 + 
c*x)]*(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x))) + ((7* 
I)*c^2*d^2*e*(c*d + e)*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x 
))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*(EllipticE[I*ArcS 
inh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] - EllipticF[I*ArcSinh 
[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)]))/(c*d - e) - ((7*I)*c*d 
*e^2*(c*d + e)*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + 
c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*(EllipticE[I*ArcSinh[Sqrt 
[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] - EllipticF[I*ArcSinh[Sqrt[(1 
 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)]))/(c*d - e) + (3*I)*c^3*d^3*Sqrt 
[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 
- c*x)/(1 + c*x)))/(c*d + e)]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x) 
]], (c*d - e)/(c*d + e)] - (2*I)*c^2*d^2*e*Sqrt[1 + (1 - c*x)/(1 + c*x)]*S 
qrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e 
)]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] - 
I*e^3*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c* 
d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x) 
/(1 + c*x)]], (c*d - e)/(c*d + e)] + ((3 + 3*I)*c^3*d^3*(-I + Sqrt[(1 -...
 

3.1.81.3 Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.89, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {6842, 634, 632, 186, 413, 412, 2185, 27, 600, 508, 327, 511, 321}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (d+e x)^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx\)

\(\Big \downarrow \) 6842

\(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {(d+e x)^{5/2}}{x \sqrt {1-c^2 x^2}}dx}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}\)

\(\Big \downarrow \) 634

\(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (d^3 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}\)

\(\Big \downarrow \) 632

\(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (d^3 \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}\)

\(\Big \downarrow \) 186

\(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-2 d^3 \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\int \frac {-x^2 e^3-3 d x e^2-3 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {2 \int \frac {e^3 \left (9 d^2 c^2+7 d e x c^2+e^2\right )}{2 \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{3 c^2 e^2}-\frac {2 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {e \int \frac {9 d^2 c^2+7 d e x c^2+e^2}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {2 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {e \left (\left (2 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+7 c^2 d \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx\right )}{3 c^2}-\frac {2 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {e \left (\left (2 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {14 c d \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c^2}-\frac {2 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {e \left (\left (2 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {14 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c^2}-\frac {2 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {e \left (-\frac {2 \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}-\frac {14 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c^2}-\frac {2 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 e}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {e \left (-\frac {2 \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {14 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c^2}-\frac {2 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 e}\)

input

Int[(d + e*x)^(3/2)*(a + b*ArcSech[c*x]),x]
 

output

(2*(d + e*x)^(5/2)*(a + b*ArcSech[c*x]))/(5*e) + (2*b*Sqrt[(1 + c*x)^(-1)] 
*Sqrt[1 + c*x]*((-2*e^2*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2])/(3*c^2) + (e*((-1 
4*c*d*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + 
e)])/Sqrt[(c*(d + e*x))/(c*d + e)] - (2*(2*c^2*d^2 + e^2)*Sqrt[(c*(d + e*x 
))/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/( 
c*Sqrt[d + e*x])))/(3*c^2) - (2*d^3*Sqrt[1 - (e*(1 - c*x))/(c*d + e)]*Elli 
pticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/Sqrt[d + e/c - 
(e*(1 - c*x))/c]))/(5*e)
 

3.1.81.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 186

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] && GtQ[(d*e - c*f)/d, 0]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 632

Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : 
> With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a]   Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - 
q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 634

Int[((c_) + (d_.)*(x_))^(n_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> 
 Simp[c^(n + 1/2)   Int[1/(x*Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] - Int[( 
1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[(c^(n + 1/2) - (c + d*x)^(n 
+ 1/2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n - 1/2, 0]
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

rule 6842

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbo 
l] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSech[c*x])/(e*(m + 1))), x] + Simp[ 
b*(Sqrt[1 + c*x]/(e*(m + 1)))*Sqrt[1/(1 + c*x)]   Int[(d + e*x)^(m + 1)/(x* 
Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
 

3.1.81.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(817\) vs. \(2(302)=604\).

Time = 12.72 (sec) , antiderivative size = 818, normalized size of antiderivative = 2.38

method	result	size
derivativedivides	\(\frac {\frac {2 a \left (e x +d \right )^{\frac {5}{2}}}{5}+2 b \left (\frac {\left (e x +d \right )^{\frac {5}{2}} \operatorname {arcsech}\left (c x \right )}{5}-\frac {2 e^{2} \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \left (\sqrt {\frac {c}{c d +e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}+9 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, c^{2} d^{2}-7 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, c^{2} d^{2}-3 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, c^{2} d^{2}-2 \sqrt {\frac {c}{c d +e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}-7 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, c d e +7 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, c d e +\sqrt {\frac {c}{c d +e}}\, c^{2} d^{2} \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, e^{2}-\sqrt {\frac {c}{c d +e}}\, e^{2} \sqrt {e x +d}\right )}{15 c \sqrt {\frac {c}{c d +e}}\, \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\)	\(818\)
default	\(\frac {\frac {2 a \left (e x +d \right )^{\frac {5}{2}}}{5}+2 b \left (\frac {\left (e x +d \right )^{\frac {5}{2}} \operatorname {arcsech}\left (c x \right )}{5}-\frac {2 e^{2} \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \left (\sqrt {\frac {c}{c d +e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}+9 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, c^{2} d^{2}-7 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, c^{2} d^{2}-3 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, c^{2} d^{2}-2 \sqrt {\frac {c}{c d +e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}-7 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, c d e +7 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, c d e +\sqrt {\frac {c}{c d +e}}\, c^{2} d^{2} \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, e^{2}-\sqrt {\frac {c}{c d +e}}\, e^{2} \sqrt {e x +d}\right )}{15 c \sqrt {\frac {c}{c d +e}}\, \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\)	\(818\)
parts	\(\frac {2 a \left (e x +d \right )^{\frac {5}{2}}}{5 e}+\frac {2 b \left (\frac {\left (e x +d \right )^{\frac {5}{2}} \operatorname {arcsech}\left (c x \right )}{5}-\frac {2 e^{2} \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c e x}}\, x \sqrt {\frac {c \left (e x +d \right )-c d +e}{c e x}}\, \left (\sqrt {\frac {c}{c d +e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}+9 \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) c^{2} d^{2}-7 \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) c^{2} d^{2}-3 \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) c^{2} d^{2}-2 \sqrt {\frac {c}{c d +e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}-7 \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) c d e +7 \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) c d e +\sqrt {\frac {c}{c d +e}}\, c^{2} d^{2} \sqrt {e x +d}+\sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) e^{2}-\sqrt {\frac {c}{c d +e}}\, e^{2} \sqrt {e x +d}\right )}{15 c \sqrt {\frac {c}{c d +e}}\, \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\)	\(832\)

input

int((e*x+d)^(3/2)*(a+b*arcsech(c*x)),x,method=_RETURNVERBOSE)
 

output

2/e*(1/5*a*(e*x+d)^(5/2)+b*(1/5*(e*x+d)^(5/2)*arcsech(c*x)-2/15/c*e^2*((-c 
*(e*x+d)+c*d+e)/c/e/x)^(1/2)*x*(-(-c*(e*x+d)+c*d-e)/c/e/x)^(1/2)*((c/(c*d+ 
e))^(1/2)*c^2*(e*x+d)^(5/2)+9*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*EllipticF 
((e*x+d)^(1/2)*(c/(c*d+e))^(1/2),((c*d+e)/(c*d-e))^(1/2))*((-c*(e*x+d)+c*d 
+e)/(c*d+e))^(1/2)*c^2*d^2-7*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*EllipticE( 
(e*x+d)^(1/2)*(c/(c*d+e))^(1/2),((c*d+e)/(c*d-e))^(1/2))*((-c*(e*x+d)+c*d+ 
e)/(c*d+e))^(1/2)*c^2*d^2-3*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*EllipticPi( 
(e*x+d)^(1/2)*(c/(c*d+e))^(1/2),1/c*(c*d+e)/d,(c/(c*d-e))^(1/2)/(c/(c*d+e) 
)^(1/2))*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*c^2*d^2-2*(c/(c*d+e))^(1/2)*c^ 
2*d*(e*x+d)^(3/2)-7*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*EllipticF((e*x+d)^( 
1/2)*(c/(c*d+e))^(1/2),((c*d+e)/(c*d-e))^(1/2))*((-c*(e*x+d)+c*d+e)/(c*d+e 
))^(1/2)*c*d*e+7*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*EllipticE((e*x+d)^(1/2 
)*(c/(c*d+e))^(1/2),((c*d+e)/(c*d-e))^(1/2))*((-c*(e*x+d)+c*d+e)/(c*d+e))^ 
(1/2)*c*d*e+(c/(c*d+e))^(1/2)*c^2*d^2*(e*x+d)^(1/2)+((-c*(e*x+d)+c*d-e)/(c 
*d-e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d+e))^(1/2),((c*d+e)/(c*d-e))^( 
1/2))*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*e^2-(c/(c*d+e))^(1/2)*e^2*(e*x+d) 
^(1/2))/(c/(c*d+e))^(1/2)/(c^2*(e*x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2-e^2)))
 

3.1.81.5 Fricas [F(-1)]

Timed out. \[ \int (d+e x)^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Timed out} \]

input

integrate((e*x+d)^(3/2)*(a+b*arcsech(c*x)),x, algorithm="fricas")
 

output

Timed out
 

3.1.81.6 Sympy [F]

\[ \int (d+e x)^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x\right )^{\frac {3}{2}}\, dx \]

input

integrate((e*x+d)**(3/2)*(a+b*asech(c*x)),x)
 

output

Integral((a + b*asech(c*x))*(d + e*x)**(3/2), x)
 

3.1.81.7 Maxima [F(-2)]

Exception generated. \[ \int (d+e x)^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]

input

integrate((e*x+d)^(3/2)*(a+b*arcsech(c*x)),x, algorithm="maxima")
 

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e+c*d>0)', see `assume?` for mor 
e details)
 

3.1.81.8 Giac [F]

\[ \int (d+e x)^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (e x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} \,d x } \]

input

integrate((e*x+d)^(3/2)*(a+b*arcsech(c*x)),x, algorithm="giac")
 

output

integrate((e*x + d)^(3/2)*(b*arcsech(c*x) + a), x)
 

3.1.81.9 Mupad [F(-1)]

Timed out. \[ \int (d+e x)^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^{3/2} \,d x \]

input

int((a + b*acosh(1/(c*x)))*(d + e*x)^(3/2),x)
 

output

int((a + b*acosh(1/(c*x)))*(d + e*x)^(3/2), x)